QUASIMINIMAL DISTAL FUNCTION SPACE AND ITS SEMIGROUP COMPACTIFICATION

Quasiminimal distal function on a semitopological semigroup is 
introduced. The concept of right topological semigroup compactification is 
employed to study the algebra of quasiminimal distal functions. The universal 
mapping property of the quasiminimal distal compactification is obtained.

In a recent paper, we generalized the notion of distal flows and distal functions on an arbitrary semitopological semigroup S and studied their function spaces [3].The study was an extension of Junghenn's work on distal functions and their semigroup compactifications [2].The impetus for the author's work in this area is the study by Berglund et al. in which the authors have introduced several kinds of function spaces which are admissible C*-subalgebras of C(S) and studied these function spaces through semigroup compactifications [i].Interestingly, these compactifications possess certain universal mapping properties, universal   in the sense that the compactifications are maximal with respect to these properties.This approach of studying function spaces through semigroup compactifications enables one to take advantage of the universal mapping properties the compactifications enjoy.For example, in the case of distal functions on a semigroup S, Junghenn has effectively used the universal mapping properties to show that the space of weakly almost periodic distal functions on S is the direct sum of the algebra of strongly almost periodic functions and two ideals of weakly almost "flight" functions on S [2].In this paper, we introduce the quasiminimal distal function space QMD(S), and show that this space is an admissible C*osubalgebra of C(S).As a consequence, we obtain the QMD(S)-compactification of S which is the spectrum of QMD(S).We then characterize this semigroup compactification in terms of the universal mapping property the compactification possesses.where the operators h and Rs on C(S) are defined by Lsf(t) f(st) and Rsf(t)   f(ts) for s, t S and f C(S).We define Tx.F B by Txf(. x(L f) where f F and x the spectrum of F which is the space of nonzero continuous complex homomorphisms on F. A C*-subalgebra F of C(S) is called admissible if it is translation invariant, contains the constant functions, and TxF=F for every x spectrum of F. A pair (X,) is called right topological Gompactification of S if X is a compact right topological semigroup and a'S X is a continuous homomorphism with dense image such that the mapping x a(s)x'X X is continuous for each s S.
It is called an F-compactification of S if F is an admissible subalgebra of C(S) and * C(X) F. We call X the phase .space of the compactification.The set of all multiplicative means on F, denoted by MM(F), is a w-compact right topological semigroup with binary operation defined in MM(F) by xy xoTy.If eS MM(F) is the evaluation map (e(s)(f) is dense in MM(F) and the pair (MM(F), e) is an F-compactification of S. For a fixed F, all Focompactifications are isomorphic both algebraically and topologically.Thus, there is a unique F-compactification of S. We call (MM(F), e), the canonical F-compactification of S. The map Tx'F B has the following properties (for details see [I. 1.4]).For s S, f F and x, y X, a) T TxoTy b) [e(s)x](f) x (Lsf) c) [x e(s)](f) x (Rsf).A right topological compactification (X,) of S is said to be maximal (universal) with respect to a property P if (X,) has the property P and if (Y,) is a right topological compactification of S with property P, then factors in the sense that %o for some continuous homomorphism %'XY.A very useful admissible subalgebra of C(S) is LMC(S) If C(S) s x (Lsf) is continuous for every x MM C(S).It is the largest admissible subalgebra of C(S) and the hMC(S)- compactification of S is maximal with respect to the property that it is a right If X is a compact topological space and 'S X X is a continuous homomorphism such that (s)'X X is continuous for each s S, then the triple (S, X, ) is called a flo____w.For convenience, we write sx for (s)(X).The flow is called distal (respectively, quasidistal) if, whenever x,y,X such that lim s x lim s y for some net (s) in S, then x y (respectively, sx sy for every s S).A function f IMC(S) is called a distal (respectively, a quasidistal) function if the flow (S, Z, ), where Z is the closure of Rsf in the topology of pointwise convergence on C(S) and (s)'flz is distal (respectively, a quasidistal).It is called a minimal function if g Z implies f Z. Junghenn has shown that the set of all minimal distal functions, MD(S), is an admissible subalgebra of C(S) and that MD(S)ocompactification (Y,) is maximal with respect to the property that Y is left simple [2].The author has proved analogous results for quasidistal functions, QD(S).For more on these functions and related theorems, the reader is referred to [3,4].

QUASIMINIMAL DISTAL FUNCTIONS DEFINITION 2-1.
A function of f LMC(S) is quasiminimal if for every g Z, there exists x X such that Tuf Tuxg for every u X where X is the phase space of the canonical LMC(S)-compactification (X,).
We denote the set of all quasiminimal functions by QM(S).X (depending on y and f) such that Tuxyf Tuf for every u X. PROOF.
Necessity" Assume f is quasiminimal.Let y X.
Then Tyf {Txf x X} Zf By definition, there exists x X such that Tuf TuxTyf Tuxyf for every X.
Sufficiency" Let g Zf {Txf x X which implies that g Tyf for some y X.For this y, there exists x X such that Tuf Tuxyf TuxTyf Txg for every X.Then by definition, f QM(S).
DEFINITION" 2"3.A function of LMC(S) is said to be quasiminimal distal if f QM(S)nQD(S).We write QMO(S) for QM(S)nQD(S).Before we present the proof of this proposition, we state here without proof the author's characterization of quasidistal functions QD(S) A function f LMC(S) is quasidistal, that is f QD(S) if and only if uev(f) uv(f) for every u 7 , v X and e E(X) where X is the phase space of the LMC(S)-compactification of S.
By Proposition i, there exists x X such that Tuxef Tuf for every u X (2.2).

Sufficiency.
It suffices to prove that f QM(S).We recall that X has the smallest ideal K(X) and for each e E (K(X)), eXe is a maximal subgroup of X with identity e [5].Let y X.There exists x X such that (exe)(eye) e which implies exeye e.
QMD(S) is one admissible subalgebra of C(S).
Linearity of the space QMD(S) is immediate from Proposition 2.4.Let f QMD(S), v Z and e E(X).For s S, ve(Lsf) (s)ve(f) --(s)v(f) (Prop.2.4)   v(Lsf) which implies that Lsf QMD(S).Now ve(Rsf vem(s)(f) v(s)(f) (Remark 2.5) v(Rsf) which proves that R_f QMD(S).Thus QMD(S) is translation invariant.That QMD(S) is an algebra follows from the fact that X is the set of all multiplicative means on LMC(S).Trivially, v(1) ve(1).So QMD(S) contains 1 and hence contains all constant functions.We complete this proof by showing that Tzf QMD(S) for every z MM QMD(S).Let z MM QMD(S) and x MM QMD(S) be the restriction map.I.e., D(x) xlQMD(S ).Then is a continuous homomorphism onto MM QMD(S).There exists w X such that D(w) z.
For s S, Tf(s) w(Lf) z(Lsf) Tf(s).Thus Tf Tzf.Let v Z, u X, i. PRELIMINARIES Throughout our work, S denotes a semitopological semigroup with Hausdorff topology and C(S) denotes the C*-algebra of all bounded complex-valued continuous functions on S. A subspace F of C(S) is translation invariant if LsF=F and RsF=F and (X,a) denote the canonical LMC(S)compactification of S. Then f QM(S) if and only if for each y X, there exists x

PROPOSITION 2 4 .
Let f LMC(S) and (X,a) denote the canonical LMC(S)compactification of S. f QMD(S) if and only if f QD(S) and v(f) ve(f) for every v f, e E(x) Remark 2-5.